When an Elliptic Curve met a Modular form - A short tale on Reciprocity Laws

Reciprocity laws constitute a very general theme in modern number theory where we try to relate one kind of object to a very different kind of object in a very specific way.
One such reciprocity law tries to relate these very special Cubic equations called Elliptic Curves to highly symmetric functions on Upper half of Complex plane called Modular forms.
The two sort of objects are related in such a way that various Invariants attached to them match up perfectly. One consequence of such a correspondence is that we have way of translating problem about first object into problem about the second object which may be easier to solve. ( Like counting number of solutions modulo p on Elliptic curves using Fourier coefficient of modular form )
The famous Shimura-Taniyama Conjecture , solution to which by Andrew Wiles lead to the proof of Fermat's Last Theorem , claims that all Elliptic Curves that exist out there have an associated modular form.
#ellipticcurves #modularforms #fermatlasttheorem #langlandsprogram #andrewwiles